Descent for Monads

Pieter Hofstra and Federico De Marchi

Motivated by a desire to gain a better understanding of
the ``dimension-by-dimension'' decompositions of certain prominent
monads in higher category theory, we investigate descent theory for
endofunctors and monads. After setting up a basic framework of indexed
monoidal categories, we describe a suitable subcategory of
Cat over which we can view the assignment C |-> Mnd(C)
as an indexed category; on this base
category, there is a natural topology. Then we single out a class of
monads which are well-behaved with respect to reindexing. The main
result is now, that such monads form a stack. Using this, we can shed
some light on the free strict $\omega$-category monad on globular sets
and the free operad-with-contraction monad on the category of
collections.